The quadratic Casimir \(C_2 = L_{\alpha \beta } L^{\alpha \beta }\)  is a second order differential operator in the above representation. Let \(4C^{(12)}_2 = -\big (L^1_{\alpha \beta }+L^2_{\alpha \beta }\big )\big (L^{1,\alpha \beta }+L^{2,\alpha \beta }\big )\) . A conformal partial wave captures the contribution to the correlation function of one conformal family present in the OPE. As noticed by Dolan and Osborn, [1]}, waves may be characterised as eigenfunctions of \(C_2^{(12)}\) . When acting on \(G_4(x_i)\) , the operator \(C_2^{(12)}\)  produces a function of the same product form (REF ). Therefore, the operator \(\Delta _2=\Omega (x_i)^{-1} C_2^{(12)} \Omega (x_i)\)  can be written as a differential operator in the cross ratios, called the Casimir differential operator. It is in fact simpler to write \(\Delta _2\)  in variables \(z_i\) , where it can be shown to take the form
\(\Delta _2 = D_{z_1} + D_{z_2} + (d-2)\frac{z_1 z_2}{z_1 - z_2}((1-z_1)\partial _{z_1} - (1-z_2)\partial _{z_2}) .\) 
